Question: Simon has $160$ meters of fencing to build a rectangular garden. The garden's area (in square meters) as a function of the garden's width $x$ (in meters) is modeled by $A(x)=-x(x-80)$ What is the maximum area possible?
Explanation: The garden's area is modeled by a quadratic function, whose graph is a parabola. The maximum area is reached at the vertex. So in order to find the maximum area, we need to find the vertex's $y$ -coordinate. We will start by finding the vertex's $x$ -coordinate, and then plug that into $A(x)$. The vertex's $x$ -coordinate is the average of the two zeros, so let's find those first. $\begin{aligned} A(x)&=0 \\\\ -x(x-80)&=0 \\\\ \swarrow &\searrow \\\\ -x=0\text{ or }&x-80=0 \\\\ x={0}\text{ or }&x={80} \end{aligned}$ Now let's take the zeros' average: $\dfrac{({0})+({80})}{2}=\dfrac{80}{2}={40}$ The vertex's $x$ -coordinate is ${40}$. Now let's find $A({{40}})$ : $\begin{aligned} A({40})&=-({40})({40}-80) \\\\ &=-(40)(-40) \\\\ &=1600 \end{aligned}$ In conclusion, the maximum area is $1600$ square meters.